Two Types Of Predator-prey Epidemic Systems With Bilinear Incidence

Infectious disease dynamics is the important quantitative analysis method for regularity of population epidemic. Through the analysis for the dynamics of predator-prey with infectious disease, we can understand and forecast the trend of population's development when illness gets popular and seek the countermeasure for the prevention and the control of disease popular. To be the question of two types of bilinear incidence of predator-prey epidemic systems, we study the final survive condition of population by making use of the differential equation stable method and the correlation theories and provide the basis for theoretical research to enable the existence of population persistently.In the first part we study the SIS epidemiological ecological system with bilinear incidence which only the predator catches an illness. This system considers that the prey population with general density restriction affects the predator population which catches an illness. We discuss the existence unique condition of nonnegative equilibrium points of the system by the qualitative analysis of ordinary differential equation. The existent condition of the positive equilibrium point is obtained by image analysis. The threshold value of the disease-free and the population extinction are found. The partial stability of the system equilibrium points is obtained by analyzing eigenvalue. We obtain the boundedness of solution of the system by the comparison principle. The global stability of the disease-free equilibrium point and the population extinct equilibrium point is discussed by constructing Liapunov function and the theory of limit equation. The condition of permanent existence of the system is obtained by the theory of permanent existence of the solutions on infinite-dimension dynamical system.In the second part we study the SIS epidemiological ecological system with bilinear incidence which both the prey and the predator catch an illness. This system considers inter-infection of disease. According to the relations of the communicator, the infectious population, prey population and predator population and the limit equation theory , this paper divides the system into two subsystems that one subsystem has no infectious population the other subsystem has infectious population. For no infectious subsystem, we obtain the existent condition of the subsystem equilibrium point, sufficient condition of partial and global stability by the differential equation stability theory. For the infective subsystem, we obtain the existent condition of the subsystem equilibrium point and the uniqueness of the positive equilibrium point by the no infectious subsystem's conclusion and the image analysis. The partial stability of the subsystem equilibrium points is discussed by analyzing eigenvalue. The condition that the system has no limit cycle is obtained by the Dulac function. We obtain the global stability of the disease-free equilibrium point and the endemic equilibrium point by limit equation theory.